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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1omo | Structured version Visualization version GIF version |
Description: There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 45803 assuming ax-un 7501 (see f1omoALT 45805). (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
f1omo.1 | ⊢ (𝜑 → 𝐹 = (𝐴 × {1o})) |
Ref | Expression |
---|---|
f1omo | ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 8193 | . . . 4 ⊢ 1o ∈ V | |
2 | eqid 2736 | . . . 4 ⊢ ((𝐴 × {1o})‘𝑋) = ((𝐴 × {1o})‘𝑋) | |
3 | 1, 2 | fvconst0ci 45802 | . . 3 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) |
4 | mo0 45775 | . . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)) | |
5 | el1o 8204 | . . . . . . . 8 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅) | |
6 | el1o 8204 | . . . . . . . 8 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅) | |
7 | eqtr3 2758 | . . . . . . . 8 ⊢ ((𝑦 = ∅ ∧ 𝑥 = ∅) → 𝑦 = 𝑥) | |
8 | 5, 6, 7 | syl2anb 601 | . . . . . . 7 ⊢ ((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥) |
9 | 8 | gen2 1804 | . . . . . 6 ⊢ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥) |
10 | eleq1w 2813 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 1o ↔ 𝑥 ∈ 1o)) | |
11 | 10 | mo4 2565 | . . . . . 6 ⊢ (∃*𝑦 𝑦 ∈ 1o ↔ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥)) |
12 | 9, 11 | mpbir 234 | . . . . 5 ⊢ ∃*𝑦 𝑦 ∈ 1o |
13 | eleq2w2 2732 | . . . . . 6 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ 𝑦 ∈ 1o)) | |
14 | 13 | mobidv 2548 | . . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ ∃*𝑦 𝑦 ∈ 1o)) |
15 | 12, 14 | mpbiri 261 | . . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)) |
16 | 4, 15 | jaoi 857 | . . 3 ⊢ ((((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)) |
17 | 3, 16 | ax-mp 5 | . 2 ⊢ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) |
18 | f1omo.1 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝐴 × {1o})) | |
19 | 18 | fveq1d 6697 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋)) |
20 | 19 | eleq2d 2816 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝑋) ↔ 𝑦 ∈ ((𝐴 × {1o})‘𝑋))) |
21 | 20 | mobidv 2548 | . 2 ⊢ (𝜑 → (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))) |
22 | 17, 21 | mpbiri 261 | 1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 ∀wal 1541 = wceq 1543 ∈ wcel 2112 ∃*wmo 2537 ∅c0 4223 {csn 4527 × cxp 5534 ‘cfv 6358 1oc1o 8173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-1o 8180 |
This theorem is referenced by: indthinc 45949 prsthinc 45951 |
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